Fit log returns to F-S skew standardized Student-t distribution.
m is the location parameter.
s is the scale parameter.
nu is the estimated shape parameter (degrees of freedom).
xi is the estimated skewness parameter.

Log returns data 2011-2023.

For 2011, medium risk data is used in the high risk data set, as no high risk fund data is available prior to 2012.
vmrl is a long version of Velliv medium risk data, from 2007 to 2023. For 2007 to 2011 (both included) no high risk data is available.

Summary of gross returns

##       vmr             pmr             mmr             vhr       
##  Min.   :0.868   Min.   :0.904   Min.   :0.988   Min.   :0.849  
##  1st Qu.:1.044   1st Qu.:1.042   1st Qu.:1.013   1st Qu.:1.039  
##  Median :1.097   Median :1.084   Median :1.085   Median :1.099  
##  Mean   :1.070   Mean   :1.065   Mean   :1.066   Mean   :1.085  
##  3rd Qu.:1.136   3rd Qu.:1.107   3rd Qu.:1.101   3rd Qu.:1.160  
##  Max.   :1.168   Max.   :1.141   Max.   :1.133   Max.   :1.214  
##       phr             mhr       
##  Min.   :0.878   Min.   :0.977  
##  1st Qu.:1.068   1st Qu.:1.013  
##  Median :1.128   Median :1.113  
##  Mean   :1.095   Mean   :1.087  
##  3rd Qu.:1.182   3rd Qu.:1.128  
##  Max.   :1.208   Max.   :1.207
##       vmrl      
##  Min.   :0.801  
##  1st Qu.:1.013  
##  Median :1.085  
##  Mean   :1.061  
##  3rd Qu.:1.128  
##  Max.   :1.193
##            vmr   pmr   mmr   vhr   phr   mhr
## Min.   : 0.868 0.904 0.988 0.849 0.878 0.977
## 1st Qu.: 1.044 1.042 1.013 1.039 1.068 1.013
## Median : 1.097 1.084 1.085 1.099 1.128 1.113
## Mean   : 1.070 1.065 1.066 1.085 1.095 1.087
## 3rd Qu.: 1.136 1.107 1.101 1.160 1.182 1.128
## Max.   : 1.168 1.141 1.133 1.214 1.208 1.207

Ranking

Min. : ranking 1st Qu.: ranking Median : ranking Mean : ranking 3rd Qu.: ranking Max. : ranking
0.988 mmr 1.068 phr 1.128 phr 1.095 phr 1.136 vmr 1.168 vmr
0.977 mhr 1.044 vmr 1.113 mhr 1.087 mhr 1.107 pmr 1.141 pmr
0.904 pmr 1.042 pmr 1.099 vhr 1.085 vhr 1.101 mmr 1.133 mmr
0.878 phr 1.039 vhr 1.097 vmr 1.070 vmr 1.160 vhr 1.214 vhr
0.868 vmr 1.013 mmr 1.085 mmr 1.066 mmr 1.182 phr 1.208 phr
0.849 vhr 1.013 mhr 1.084 pmr 1.065 pmr 1.128 mhr 1.207 mhr

Covariance

## cov(vmr, pmr) =  -0.001094875
## cov(vhr, phr) =  -0.0001730651

Velliv medium risk, 2011 - 2023

## 
## AIC: -27.8497 
## BIC: -25.58991 
## m: 0.0480931 
## s: 0.1198426 
## nu (df): 3.303595 
## xi: 0.03361192 
## R^2: 0.993 
## 
## An R^2 of 0.993 suggests that the fit is extremely good.
## 
## What is the risk of losing max 10 %? =< 7.4 percent
## What is the risk of losing max 25 %? =< 1.8 percent
## What is the risk of losing max 50 %? =< 0.2 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
## 
## What is the chance of gaining min 10 %? >= 41 percent
## What is the chance of gaining min 25 %? >= 0 percent
## What is the chance of gaining min 50 %? >= 0 percent
## What is the chance of gaining min 90 %? >= 0 percent
## What is the chance of gaining min 99 %? >= 0 percent

QQ Plot

The qq plot looks great. Log returns for Velliv medium risk seems to be consistent with a skewed t-distribution.

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

We see that for a few observations out of a 1000, the losses are disastrous, while the upside is very dampened.

Monte Carlo

## Down-and-out simulation:
## Probability of down-and-out: 0 percent
## 
## Mean portfolio index value after 20 years: 280.919 kr.
## SD of portfolio index value after 20 years: 124.661 kr.
## Min total portfolio index value after 20 years: 0.145 kr.
## Max total portfolio index value after 20 years: 932.163 kr.
## 
## Share of paths finishing below 100: 4.65 percent

Importance sampling

Max vs sum plots

Max vs sum plots for the first four moments:

Velliv medium risk, 2007 - 2023

Fit to skew t distribution

## 
## AIC: -34.35752 
## BIC: -31.02467 
## m: 0.05171176 
## s: 0.1149408 
## nu (df): 2.706099 
## xi: 0.5049945 
## R^2: 0.978 
## 
## An R^2 of 0.978 suggests that the fit is very good.
## 
## What is the risk of losing max 10 %? =< 5.4 percent
## What is the risk of losing max 25 %? =< 1.3 percent
## What is the risk of losing max 50 %? =< 0.2 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
## 
## What is the chance of gaining min 10 %? >= 36.2 percent
## What is the chance of gaining min 25 %? >= 0.3 percent
## What is the chance of gaining min 50 %? >= 0 percent
## What is the chance of gaining min 90 %? >= 0 percent
## What is the chance of gaining min 99 %? >= 0 percent

QQ Plot

The qq plot looks good. Log returns for Velliv high risk seems to be consistent with a skewed t-distribution.

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

We see that for a few observations out of a 1000, the losses are disastrous, while the upside is very dampened. But because the disastrous loss in 2008 was followed by a large profit the following year, we see some increased upside for the top percentiles. Beware: A 1.2 return following a 0.8 return doesn’t take us back where we were before the loss. Path dependency! So if returns more or less average out, but high returns have a tendency to follow high losses, that’s bad!

Monte Carlo

## Down-and-out simulation:
## Probability of down-and-out: 0 percent
## 
## Mean portfolio index value after 20 years: 294.822 kr.
## SD of portfolio index value after 20 years: 120.757 kr.
## Min total portfolio index value after 20 years: 0.049 kr.
## Max total portfolio index value after 20 years: 2891.229 kr.
## 
## Share of paths finishing below 100: 3.3 percent

Max vs sum plots

Max vs sum plots for the first four moments:

Velliv high risk, 2011 - 2023

Fit to skew t distribution

## 
## AIC: -21.42488 
## BIC: -19.16508 
## m: 0.06471454 
## s: 0.1499924 
## nu (df): 3.144355 
## xi: 0.002367034 
## R^2: 0.991 
## 
## An R^2 of 0.991 suggests that the fit is extremely good.
## 
## What is the risk of losing max 10 %? =< 8.3 percent
## What is the risk of losing max 25 %? =< 2.5 percent
## What is the risk of losing max 50 %? =< 0.4 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
## 
## What is the chance of gaining min 10 %? >= 53.3 percent
## What is the chance of gaining min 25 %? >= 0 percent
## What is the chance of gaining min 50 %? >= 0 percent
## What is the chance of gaining min 90 %? >= 0 percent
## What is the chance of gaining min 99 %? >= 0 percent

QQ Plot

The qq plot looks great. Returns for Velliv medium risk seems to be consistent with a skewed t-distribution.

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

We see that for a few observations out of a 1000, the losses are disastrous, while the upside is very dampened.

Monte Carlo

## Down-and-out simulation:
## Probability of down-and-out: 0 percent
## 
## Mean portfolio index value after 20 years: 405.123 kr.
## SD of portfolio index value after 20 years: 214.867 kr.
## Min total portfolio index value after 20 years: 0.832 kr.
## Max total portfolio index value after 20 years: 1599.122 kr.
## 
## Share of paths finishing below 100: 4.11 percent

Max vs sum plots

Max vs sum plots for the first four moments:

PFA medium risk, 2011 - 2023

Fit to skew t distribution

## 
## AIC: -33.22998 
## BIC: -30.97018 
## m: 0.05789224 
## s: 0.1234592 
## nu (df): 2.265273 
## xi: 0.477324 
## R^2: 0.991 
## 
## An R^2 of 0.991 suggests that the fit is extremely good.
## 
## What is the risk of losing max 10 %? =< 3.3 percent
## What is the risk of losing max 25 %? =< 0.9 percent
## What is the risk of losing max 50 %? =< 0.2 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
## 
## What is the chance of gaining min 10 %? >= 32.7 percent
## What is the chance of gaining min 25 %? >= 0.1 percent
## What is the chance of gaining min 50 %? >= 0 percent
## What is the chance of gaining min 90 %? >= 0 percent
## What is the chance of gaining min 99 %? >= 0 percent

QQ Plot

The qq plot looks great. Log returns for PFA medium risk seems to be consistent with a skewed t-distribution.

##  [1] -0.091256521 -0.003731241  0.027312079  0.045808232  0.059068633
##  [6]  0.069575113  0.078454727  0.086316936  0.093536451  0.100370932
## [11]  0.107018607  0.114081432  0.127604387

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

We see that for a few observations out of a 1000, the losses are disastrous. While there is some uptick at the top percentiles, the curve basically flattens out.

Monte Carlo

## Down-and-out simulation:
## Probability of down-and-out: 0 percent
## 
## Mean portfolio index value after 20 years: 322.345 kr.
## SD of portfolio index value after 20 years: 105.707 kr.
## Min total portfolio index value after 20 years: 0.018 kr.
## Max total portfolio index value after 20 years: 966.035 kr.
## 
## Share of paths finishing below 100: 2.11 percent

Max vs sum plots

Max vs sum plots for the first four moments:

PFA high risk, 2011 - 2023

Fit to skew t distribution

## 
## AIC: -23.72565 
## BIC: -21.46585 
## m: 0.08386034 
## s: 0.1210107 
## nu (df): 3.184569 
## xi: 0.01790306 
## R^2: 0.964 
## 
## An R^2 of 0.964 suggests that the fit is very good.
## 
## What is the risk of losing max 10 %? =< 5.3 percent
## What is the risk of losing max 25 %? =< 1.4 percent
## What is the risk of losing max 50 %? =< 0.2 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
## 
## What is the chance of gaining min 10 %? >= 59.6 percent
## What is the chance of gaining min 25 %? >= 0 percent
## What is the chance of gaining min 50 %? >= 0 percent
## What is the chance of gaining min 90 %? >= 0 percent
## What is the chance of gaining min 99 %? >= 0 percent

QQ Plot

The qq plot looks ok. Returns for PFA high risk seems to be consistent with a skewed t-distribution.

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

We see that for a few observations out of a 1000, the losses are disastrous, while the upside is very dampened.

Monte Carlo

## Down-and-out simulation:
## Probability of down-and-out: 0 percent
## 
## Mean portfolio index value after 20 years: 551.326 kr.
## SD of portfolio index value after 20 years: 242.985 kr.
## Min total portfolio index value after 20 years: 5.49 kr.
## Max total portfolio index value after 20 years: 1717.134 kr.
## 
## Share of paths finishing below 100: 0.92 percent

Max vs sum plots

Max vs sum plots for the first four moments:

Mix medium risk, 2011 - 2023

Fit to skew t distribution

## 
## AIC: -36.9603 
## BIC: -34.7005 
## m: 0.05902873 
## s: 0.08757749 
## nu (df): 2.772621 
## xi: 0.02904471 
## R^2: 0.89 
## 
## An R^2 of 0.89 suggests that the fit is not completely random.
## 
## What is the risk of losing max 10 %? =< 3.3 percent
## What is the risk of losing max 25 %? =< 0.7 percent
## What is the risk of losing max 50 %? =< 0.1 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
## 
## What is the chance of gaining min 10 %? >= 35.6 percent
## What is the chance of gaining min 25 %? >= 0 percent
## What is the chance of gaining min 50 %? >= 0 percent
## What is the chance of gaining min 90 %? >= 0 percent
## What is the chance of gaining min 99 %? >= 0 percent

QQ Plot

The fit suggests big losses for the lowest percentiles, which are not present in the data.
So the fit is actually a very cautious estimate.

Data vs fit

Let’s plot the fit and the observed returns together.

Interestingly, the fit predicts a much bigger “biggest loss” than the actual data. This is the main reason that R^2 is 0.90 and not higher.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

We see that for a few observations out of a 1000, the losses are disastrous, while the upside is very dampened.

Monte Carlo

Version a: Simulation from estimated distribution of returns of mix.

## Down-and-out simulation:
## Probability of down-and-out: 0 percent
## 
## Mean portfolio index value after 20 years: 326.089 kr.
## SD of portfolio index value after 20 years: 98.683 kr.
## Min total portfolio index value after 20 years: 2.068 kr.
## Max total portfolio index value after 20 years: 705.465 kr.
## 
## Share of paths finishing below 100: 1.07 percent

Version b: Mix of simulations from estimated distribution of returns from individual funds.

## Down-and-out simulation:
## Probability of down-and-out: 0 percent
## 
## Mean portfolio index value after 20 years: 301.426 kr.
## SD of portfolio index value after 20 years: 101.287 kr.
## Min total portfolio index value after 20 years: 34.753 kr.
## Max total portfolio index value after 20 years: 6352.682 kr.
## 
## Share of paths finishing below 100: 0.34 percent

Max vs sum plots

Max vs sum plots for the first four moments:

Mix high risk, 2011 - 2023

Fit to skew t distribution

## 
## AIC: -24.26084 
## BIC: -22.00104 
## m: 0.0822419 
## s: 0.07129843 
## nu (df): 89.86289 
## xi: 0.7697502 
## R^2: 0.961 
## 
## An R^2 of 0.961 suggests that the fit is very good.
## 
## What is the risk of losing max 10 %? =< 0.9 percent
## What is the risk of losing max 25 %? =< 0 percent
## What is the risk of losing max 50 %? =< 0 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
## 
## What is the chance of gaining min 10 %? >= 46.1 percent
## What is the chance of gaining min 25 %? >= 1.2 percent
## What is the chance of gaining min 50 %? >= 0 percent
## What is the chance of gaining min 90 %? >= 0 percent
## What is the chance of gaining min 99 %? >= 0 percent

QQ Plot

The qq plot looks good Returns for mixed medium risk portfolios seems to be consistent with a skewed t-distribution.

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

We see that the high risk mix provides a much better upside and smaller downside.

Monte Carlo

Version a: Simulation from estimated distribution of returns of mix.

## Down-and-out simulation:
## Probability of down-and-out: 0 percent
## 
## Mean portfolio index value after 20 years: 502.954 kr.
## SD of portfolio index value after 20 years: 158.024 kr.
## Min total portfolio index value after 20 years: 115.175 kr.
## Max total portfolio index value after 20 years: 1595.851 kr.
## 
## Share of paths finishing below 100: 0 percent

Version b: Mix of simulations from estimated distribution of returns from individual funds.

## Down-and-out simulation:
## Probability of down-and-out: 0 percent
## 
## Mean portfolio index value after 20 years: 478.001 kr.
## SD of portfolio index value after 20 years: 161.788 kr.
## Min total portfolio index value after 20 years: 40.871 kr.
## Max total portfolio index value after 20 years: 1245.094 kr.
## 
## Share of paths finishing below 100: 0.21 percent

Many simulations

1e6 paths:

# Down-and-out simulation:
# Probability of down-and-out: 0 percent
# 
# Mean portfolio index value after 20 years: 478.339 kr.
# SD of portfolio index value after 20 years: 163.093 kr.
# Min total portfolio index value after 20 years: 2.233 kr.
# Max total portfolio index value after 20 years: 1561.965 kr.
# 
# Share of paths finishing below 100: 0.1181 percent

Max vs sum plots

Max vs sum plots for the first four moments:

Compare pension plans

Risk of max loss

Risk of max loss of x percent for a single period (year).
x values are row names.

Velliv_m Velliv_m_l Velliv_h PFA_m PFA_h mix_m mix_h
0 21.3 18.2 19.9 12.2 14.3 12.7 13.0
5 12.5 9.6 12.8 6.0 8.6 6.2 4.2
10 7.4 5.4 8.3 3.3 5.3 3.3 0.9
25 1.8 1.3 2.5 0.9 1.4 0.7 0.0
50 0.2 0.2 0.4 0.2 0.2 0.1 0.0
90 0.0 0.0 0.0 0.0 0.0 0.0 0.0
99 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Worst ranking for loss percentiles

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 90 ranking 99 ranking
21.3 Velliv_m 12.8 Velliv_h 8.3 Velliv_h 2.5 Velliv_h 0.4 Velliv_h 0 Velliv_m 0 Velliv_m
19.9 Velliv_h 12.5 Velliv_m 7.4 Velliv_m 1.8 Velliv_m 0.2 Velliv_m 0 Velliv_m_l 0 Velliv_m_l
18.2 Velliv_m_l 9.6 Velliv_m_l 5.4 Velliv_m_l 1.4 PFA_h 0.2 Velliv_m_l 0 Velliv_h 0 Velliv_h
14.3 PFA_h 8.6 PFA_h 5.3 PFA_h 1.3 Velliv_m_l 0.2 PFA_m 0 PFA_m 0 PFA_m
13.0 mix_h 6.2 mix_m 3.3 PFA_m 0.9 PFA_m 0.2 PFA_h 0 PFA_h 0 PFA_h
12.7 mix_m 6.0 PFA_m 3.3 mix_m 0.7 mix_m 0.1 mix_m 0 mix_m 0 mix_m
12.2 PFA_m 4.2 mix_h 0.9 mix_h 0.0 mix_h 0.0 mix_h 0 mix_h 0 mix_h

Chance of min gains

Chance of min gains of x percent for a single period (year).
x values are row names.

Velliv_m Velliv_m_l Velliv_h PFA_m PFA_h mix_m mix_h
0 78.7 81.8 80.1 87.8 85.7 87.3 87.0
5 63.8 64.9 69.2 71.5 75.8 71.4 69.9
10 41.0 36.2 53.3 32.7 59.6 35.6 46.1
25 0.0 0.3 0.0 0.1 0.0 0.0 1.2
50 0.0 0.0 0.0 0.0 0.0 0.0 0.0
100 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Best ranking for gains percentiles

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 100 ranking
87.8 PFA_m 75.8 PFA_h 59.6 PFA_h 1.2 mix_h 0 Velliv_m 0 Velliv_m
87.3 mix_m 71.5 PFA_m 53.3 Velliv_h 0.3 Velliv_m_l 0 Velliv_m_l 0 Velliv_m_l
87.0 mix_h 71.4 mix_m 46.1 mix_h 0.1 PFA_m 0 Velliv_h 0 Velliv_h
85.7 PFA_h 69.9 mix_h 41.0 Velliv_m 0.0 Velliv_m 0 PFA_m 0 PFA_m
81.8 Velliv_m_l 69.2 Velliv_h 36.2 Velliv_m_l 0.0 Velliv_h 0 PFA_h 0 PFA_h
80.1 Velliv_h 64.9 Velliv_m_l 35.6 mix_m 0.0 PFA_h 0 mix_m 0 mix_m
78.7 Velliv_m 63.8 Velliv_m 32.7 PFA_m 0.0 mix_m 0 mix_h 0 mix_h

MC risk percentiles

Risk of loss from first to last period.

_a is simulation from estimated distribution of returns of mix.
_b is mix of simulations from estimated distribution of returns from individual funds.

_m is medium.
_h is high.

Velliv_m Velliv_m_l Velliv_h PFA_m PFA_h mix_m_a mix_h_a mix_m_b mix_h_b
0 4.65 3.30 4.11 2.11 0.92 1.07 0 0.34 0.21
5 4.15 2.87 3.63 1.87 0.80 0.98 0 0.27 0.17
10 3.63 2.49 3.27 1.64 0.69 0.91 0 0.23 0.10
25 2.17 1.56 2.19 1.27 0.48 0.64 0 0.13 0.04
50 0.84 0.65 1.07 0.72 0.28 0.30 0 0.03 0.01
90 0.12 0.11 0.10 0.18 0.04 0.04 0 0.00 0.00
99 0.01 0.04 0.01 0.06 0.00 0.00 0 0.00 0.00

1e6 simulation paths of mhr_b:

0 5 10 25 50 90 99
prob_pct 0.118 0.095 0.076 0.036 0.008 0 0

Worst ranking for MC loss percentiles

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 90 ranking 99 ranking
4.65 Velliv_m 4.15 Velliv_m 3.63 Velliv_m 2.19 Velliv_h 1.07 Velliv_h 0.18 PFA_m 0.06 PFA_m
4.11 Velliv_h 3.63 Velliv_h 3.27 Velliv_h 2.17 Velliv_m 0.84 Velliv_m 0.12 Velliv_m 0.04 Velliv_m_l
3.30 Velliv_m_l 2.87 Velliv_m_l 2.49 Velliv_m_l 1.56 Velliv_m_l 0.72 PFA_m 0.11 Velliv_m_l 0.01 Velliv_m
2.11 PFA_m 1.87 PFA_m 1.64 PFA_m 1.27 PFA_m 0.65 Velliv_m_l 0.10 Velliv_h 0.01 Velliv_h
1.07 mix_m_a 0.98 mix_m_a 0.91 mix_m_a 0.64 mix_m_a 0.30 mix_m_a 0.04 PFA_h 0.00 PFA_h
0.92 PFA_h 0.80 PFA_h 0.69 PFA_h 0.48 PFA_h 0.28 PFA_h 0.04 mix_m_a 0.00 mix_m_a
0.34 mix_m_b 0.27 mix_m_b 0.23 mix_m_b 0.13 mix_m_b 0.03 mix_m_b 0.00 mix_h_a 0.00 mix_h_a
0.21 mix_h_b 0.17 mix_h_b 0.10 mix_h_b 0.04 mix_h_b 0.01 mix_h_b 0.00 mix_m_b 0.00 mix_m_b
0.00 mix_h_a 0.00 mix_h_a 0.00 mix_h_a 0.00 mix_h_a 0.00 mix_h_a 0.00 mix_h_b 0.00 mix_h_b

MC gains percentiles

Chance of gains from first to last period.
_a is simulation from estimated distribution of returns of mix.
_b is mix of simulations from estimated distribution of returns from individual funds.

Velliv_m Velliv_m_l Velliv_h PFA_m PFA_h mix_m_a mix_h_a mix_m_b mix_h_b
0 95.35 96.70 95.89 97.89 99.08 98.93 100.00 99.66 99.79
5 94.60 96.13 95.51 97.71 99.03 98.78 100.00 99.59 99.76
10 93.73 95.67 94.99 97.57 98.98 98.64 100.00 99.54 99.69
25 91.31 93.68 93.59 96.81 98.53 98.07 99.99 99.11 99.56
50 86.06 90.25 90.68 94.91 97.68 96.43 99.96 97.81 99.15
100 72.14 78.33 83.83 87.98 95.15 89.70 99.74 90.05 97.20
200 39.50 45.20 64.69 58.65 85.35 60.42 93.42 48.32 87.32
300 16.60 17.95 44.77 22.56 70.97 22.89 72.21 11.30 66.47
400 5.44 5.02 29.06 4.58 54.00 3.73 45.07 1.31 41.37
500 1.45 1.06 17.15 0.60 37.96 0.32 23.84 0.09 21.70
1000 0.00 0.02 0.51 0.00 2.51 0.00 0.29 0.01 0.09

1e6 simulation paths of mhr_b:

0 5 10 25 50 100 200 300 400 500 1000
prob 99.882 99.854 99.824 99.686 99.301 97.513 86.912 65.992 41.486 21.693 0.086

Best ranking for MC gains percentiles

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 100 ranking
100.00 mix_h_a 100.00 mix_h_a 100.00 mix_h_a 99.99 mix_h_a 99.96 mix_h_a 99.74 mix_h_a
99.79 mix_h_b 99.76 mix_h_b 99.69 mix_h_b 99.56 mix_h_b 99.15 mix_h_b 97.20 mix_h_b
99.66 mix_m_b 99.59 mix_m_b 99.54 mix_m_b 99.11 mix_m_b 97.81 mix_m_b 95.15 PFA_h
99.08 PFA_h 99.03 PFA_h 98.98 PFA_h 98.53 PFA_h 97.68 PFA_h 90.05 mix_m_b
98.93 mix_m_a 98.78 mix_m_a 98.64 mix_m_a 98.07 mix_m_a 96.43 mix_m_a 89.70 mix_m_a
97.89 PFA_m 97.71 PFA_m 97.57 PFA_m 96.81 PFA_m 94.91 PFA_m 87.98 PFA_m
96.70 Velliv_m_l 96.13 Velliv_m_l 95.67 Velliv_m_l 93.68 Velliv_m_l 90.68 Velliv_h 83.83 Velliv_h
95.89 Velliv_h 95.51 Velliv_h 94.99 Velliv_h 93.59 Velliv_h 90.25 Velliv_m_l 78.33 Velliv_m_l
95.35 Velliv_m 94.60 Velliv_m 93.73 Velliv_m 91.31 Velliv_m 86.06 Velliv_m 72.14 Velliv_m
200 ranking 300 ranking 400 ranking 500 ranking 1000 ranking
93.42 mix_h_a 72.21 mix_h_a 54.00 PFA_h 37.96 PFA_h 2.51 PFA_h
87.32 mix_h_b 70.97 PFA_h 45.07 mix_h_a 23.84 mix_h_a 0.51 Velliv_h
85.35 PFA_h 66.47 mix_h_b 41.37 mix_h_b 21.70 mix_h_b 0.29 mix_h_a
64.69 Velliv_h 44.77 Velliv_h 29.06 Velliv_h 17.15 Velliv_h 0.09 mix_h_b
60.42 mix_m_a 22.89 mix_m_a 5.44 Velliv_m 1.45 Velliv_m 0.02 Velliv_m_l
58.65 PFA_m 22.56 PFA_m 5.02 Velliv_m_l 1.06 Velliv_m_l 0.01 mix_m_b
48.32 mix_m_b 17.95 Velliv_m_l 4.58 PFA_m 0.60 PFA_m 0.00 Velliv_m
45.20 Velliv_m_l 16.60 Velliv_m 3.73 mix_m_a 0.32 mix_m_a 0.00 PFA_m
39.50 Velliv_m 11.30 mix_m_b 1.31 mix_m_b 0.09 mix_m_b 0.00 mix_m_a

Summary statistics

Fit summary

Summary for fit of log returns to an F-S skew standardized Student-t distribution.
m is the location parameter.
s is the scale parameter.
nu is the estimated degrees of freedom, or shape parameter.
xi is the estimated skewness parameter.

Velliv_medium Velliv_medium_long Velliv_high PFA_medium PFA_high mix_medium mix_high
m 0.048 0.052 0.065 0.058 0.084 0.059 0.082
s 0.120 0.115 0.150 0.123 0.121 0.088 0.071
nu 3.304 2.706 3.144 2.265 3.185 2.773 89.863
xi 0.034 0.505 0.002 0.477 0.018 0.029 0.770
R-squared 0.993 0.978 0.991 0.991 0.964 0.890 0.961

Fit statistics ranking

m ranking s ranking R-squared ranking
0.084 PFA_high 0.071 mix_high 0.993 Velliv_medium
0.082 mix_high 0.088 mix_medium 0.991 Velliv_high
0.065 Velliv_high 0.115 Velliv_medium_long 0.991 PFA_medium
0.059 mix_medium 0.120 Velliv_medium 0.978 Velliv_medium_long
0.058 PFA_medium 0.121 PFA_high 0.964 PFA_high
0.052 Velliv_medium_long 0.123 PFA_medium 0.961 mix_high
0.048 Velliv_medium 0.150 Velliv_high 0.890 mix_medium

Monte Carlo simulations summary

Monte Carlo simulations of portfolio index values (currency values).
Statistics are given for the final state of all paths.
Probability of down-and_out is calculated as the share of paths that reach 0 at some point. All subsequent values for a path are set to 0, if the path reaches at any point.
0 is defined as any value below a threshold.
losing_prob_pct is the probability of losing money. This is calculated as the share of paths finishing below index 100.

## Number of paths: 10000
Velliv_m Velliv_m_l Velliv_h PFA_m PFA_h mix_m_a mix_m_b mix_h_a mix_h_b
mc_m 280.919 294.822 405.123 322.345 551.326 326.089 301.426 502.954 478.001
mc_s 124.661 120.757 214.867 105.707 242.985 98.683 101.287 158.024 161.788
mc_min 0.145 0.049 0.832 0.018 5.490 2.068 34.753 115.175 40.871
mc_max 932.163 2891.229 1599.122 966.035 1717.134 705.465 6352.682 1595.851 1245.094
dao_pct 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
losing_pct 4.650 3.300 4.110 2.110 0.920 1.070 0.340 0.000 0.210

Ranking

mc_m ranking mc_s ranking mc_min ranking mc_max ranking dao_pct ranking losing_pct ranking
551.326 PFA_h 98.683 mix_m_a 115.175 mix_h_a 6352.682 mix_m_b 0 Velliv_m 0.00 mix_h_a
502.954 mix_h_a 101.287 mix_m_b 40.871 mix_h_b 2891.229 Velliv_m_l 0 Velliv_m_l 0.21 mix_h_b
478.001 mix_h_b 105.707 PFA_m 34.753 mix_m_b 1717.134 PFA_h 0 Velliv_h 0.34 mix_m_b
405.123 Velliv_h 120.757 Velliv_m_l 5.490 PFA_h 1599.122 Velliv_h 0 PFA_m 0.92 PFA_h
326.089 mix_m_a 124.661 Velliv_m 2.068 mix_m_a 1595.851 mix_h_a 0 PFA_h 1.07 mix_m_a
322.345 PFA_m 158.024 mix_h_a 0.832 Velliv_h 1245.094 mix_h_b 0 mix_m_a 2.11 PFA_m
301.426 mix_m_b 161.788 mix_h_b 0.145 Velliv_m 966.035 PFA_m 0 mix_m_b 3.30 Velliv_m_l
294.822 Velliv_m_l 214.867 Velliv_h 0.049 Velliv_m_l 932.163 Velliv_m 0 mix_h_a 4.11 Velliv_h
280.919 Velliv_m 242.985 PFA_h 0.018 PFA_m 705.465 mix_m_a 0 mix_h_b 4.65 Velliv_m

Comments

(Ignoring mhr_a…)

mhr has some nice properties:
- It has a relatively high nu value of 90, which means it is tending more towards exponential tails than polynomial tails. All other funds have nu values close to 3, except phr which is even worse at close to 2. (Note that for a Gaussian, nu is infinite.)
- It has the lowest losing percentage of all simulations, which is better than 1/6 that of phr.
- It has a DAO percentage of 0, which is the same as mmr, and less than phr.
- Only phr has a higher mc_m.
- It has a smaller mc_s than the individual components, vhr and phr.
- It has the highest xi of all fits, suggesting less left skewness. Density plots for vmr, phr and mmr have an extremely sharp drop, as if an upward limiter has been applied, which corresponds to extremely low xi values. The density plot for mhr is by far the most symmetrical of all the fits.
- Only mmr has as higher mc_min. However, that of mmr is 18 times higher with 62, so mmr is a clear winner here.
- Naturally, it has a mc_max smaller than the individual components, vhr and phr, but ca. 1.5 times higher then mmr.
- All the first 4 moments converge nicely. For all other fits, the 4th moment doesn’t seem to converge.

Taleb, Statistical Consequences Of Fat Tails, p. 97:
“the variance of a finite variance random variable with tail exponent \(< 4\) will be infinite”.

And p. 363:
“The hedging errors for an option portfolio (under a daily revision regime) over 3000 days, un- der a constant volatility Student T with tail exponent \(\alpha = 3\). Technically the errors should not converge in finite time as their distribution has infinite variance.”

Appendix

Average of returns vs returns of average

Math

\[\text{Avg. of returns} := \dfrac{ \left(\dfrac{x_t}{x_{t-1}} + \dfrac{y_t}{y_{t-1}} \right) }{2}\] \[\text{Returns of avg.} := \left(\dfrac{ x_t + y_t }{2}\right) \Big/ \left(\dfrac{ x_{t-1} + y_{t-1} }{2}\right) \equiv \dfrac{ x_t + y_t }{ x_{t-1} + y_{t-1}}\]

For which \(x_1\) and \(y_1\) are \(\text{Avg. of returns} = \text{Returns of avg.}\)?

\[\dfrac{ \left(\dfrac{x_t}{x_{t-1}} + \dfrac{y_t}{y_{t-1}} \right) }{2} = \dfrac{ x_t + y_t }{ x_{t-1} + y_{t-1}}\]

\[\dfrac{x_t}{x_{t-1}} + \dfrac{y_t}{y_{t-1}} = 2 \dfrac{ x_t + y_t }{ x_{t-1} + y_{t-1}}\]

\[(x_{t-1} + y_{t-1}) x_t y_{t-1} + (x_{t-1} + y_{t-1}) x_{t-1} y_t = 2 (x_{t-1}y_{t-1}x_t + x_{t-1}y_{t-1}y_t)\]

\[(x_{t-1}x_1y_{t-1} + y_{t-1}x_ty_{t-1}) + (x_{t-1}x_{t-1}y_t + x_{t-1}y_{t-1}y_t) = 2(x_{t-1}y_{t-1}x_t + x_{t-1}y_{t-1}y_t)\] This is not generally true, but true if for instance \(x_{t-1} = y_{t-1}\).

Example

Definition: R = 1+r

## Let x_0 be 100.
## Let y_0 be 200.
## So the initial value of the pf is 300 .
## Let R_x be 0.5.
## Let R_y be 1.5.

Then,

## x_1 is R_x * x_0 = 50.
## y_1 is R_y * y_0 = 300.

Average of returns:

## 0.5 * (R_x + R_y) = 1

So here the value of the pf at t=1 should be unchanged from t=0:

## (x_0 + y_0) * 0.5 * (R_x + R_y) = 300

But this is clearly not the case:

## 0.5 * (x_1 + y_1) = 0.5 * (R_x * x_0 + R_y * y_0) = 175

Therefore we should take returns of average, not average of returns!

Let’s take the average of log returns instead:

## 0.5 * (log(R_x) + log(R_y)) = -0.143841

We now get:

## (x_0 + y_0) * exp(0.5 * (log(Rx) + log(Ry))) = 259.8076

So taking the average of log returns doesn’t work either.

Simulation of mix vs mix of simulations

Test if a simulation of a mix (average) of two returns series has the same distribution as a mix of two simulated returns series.

## m(data_x): 0.05352916 
## s(data_x): 0.3079141 
## m(data_y): 11.11783 
## s(data_y): 3.918513 
## 
## m(data_x + data_y): 5.585677 
## s(data_x + data_y): 1.969611

m and s of final state of all paths.
_a is mix of simulated returns.
_b is simulated mixed returns.

m_a m_b s_a s_b
111.509 111.875 8.651 8.637
111.281 111.721 8.967 8.755
111.885 111.422 8.780 8.476
112.123 112.267 8.961 9.058
111.659 111.936 8.847 9.113
111.410 112.316 9.000 8.785
111.800 111.640 8.671 8.699
111.609 112.120 8.847 8.314
111.576 111.953 8.710 8.781
111.152 111.765 8.780 8.673 
##       m_a             m_b             s_a             s_b       
##  Min.   :111.2   Min.   :111.4   Min.   :8.651   Min.   :8.314  
##  1st Qu.:111.4   1st Qu.:111.7   1st Qu.:8.727   1st Qu.:8.646  
##  Median :111.6   Median :111.9   Median :8.813   Median :8.727  
##  Mean   :111.6   Mean   :111.9   Mean   :8.821   Mean   :8.729  
##  3rd Qu.:111.8   3rd Qu.:112.1   3rd Qu.:8.932   3rd Qu.:8.784  
##  Max.   :112.1   Max.   :112.3   Max.   :9.000   Max.   :9.113

_a and _b are very close to equal.
We attribute the differences to differences in estimating the distributions in version a and b.

The final state is independent of the order of the preceding steps:

So does the order of the steps in the two processes matter, when mixing simulated returns?

The order of steps in the individual paths do not matter, because the mix of simulated paths is a sum of a sum, so the order of terms doesn’t affect the sum. If there is variation it is because the sets preceding steps are not the same. For instance, the steps between step 1 and 60 in the plot above are not the same for the two lines.

Recall, \[\text{Var}(aX+bY) = a^2 \text{Var}(X) + b^2 \text{Var}(Y) + 2ab \text{Cov}(a, b)\]

var(0.5 * vhr + 0.5 * phr)
## [1] 0.005355618
0.5^2 * var(vhr) + 0.5^2 * var(phr) + 2 * 0.5 * 0.5 * cov(vhr, phr)
## [1] 0.005355618

Our distribution estimate is based on 13 observations. Is that enough for a robust estimate? What if we suddenly hit a year like 2008? How would that affect our estimate?
Let’s try to include the Velliv data from 2007-2010.
We do this by sampling 13 observations from vmrl.

##        m                 s          
##  Min.   :0.05922   Min.   :0.04481  
##  1st Qu.:0.06592   1st Qu.:0.05899  
##  Median :0.06865   Median :0.06483  
##  Mean   :0.07028   Mean   :0.06672  
##  3rd Qu.:0.07309   3rd Qu.:0.07481  
##  Max.   :0.08376   Max.   :0.09411

The meaning of xi

The fit for mhr has the highest xi value of all. This suggests right-skew:

Max vs sum plot

If the Law Of Large Numbers holds true, \[\dfrac{\max (X_1^p, ..., X^p)}{\sum_{i=1}^n X_i^p} \rightarrow 0\] for \(n \rightarrow \infty\).

If not, \(X\) doesn’t have a \(p\)’th moment.

See Taleb: The Statistical Consequences Of Fat Tails, p. 192